The deal marks an exit for NVM Private Equity, which first backed the business in July 2014.
LDC’s stake has not been revealed, although MSQ co-founder and CEO Peter Reid told PRWeek the management team "still has a majority voting stake".
The deal values the marcomms group - which employs more than 600 people across 15 offices in the UK, Asia and US - at around £37.5m.
LDC said the deal "will enable the group to accelerate the rollout of its successful multi-disciplinary model beyond London".
This model, which sees the different disciplines across the MSQ agencies working together, currently accounts for 75 per cent business.
MSQ will also look to invest further in "talent and services to expand and develop the group’s capabilities", adding: "This will be organically and potentially through acquisition."
Reid told PRWeek there are "a few things at the early stage" regarding potential acquisitions but said there are "no specific targets".
"We are certainly not going on a mad acquisitions trail," he stated, "It’s much more [about] continuing to build on and develop a strategy that’s been so successful over the last couple of years and look to accelerate that rather than any departure from the past."
He added: "We’ve always had an exclusively new business and organic growth-focused strategy, and really the partnership with LDC is very much continuing that theme and developing it.
"But on the back of that we have the ability to make at least modestly-sized acquisitions if it enables us to where we want to get to faster."
Reid said developing MSQ’s New York hub, and potentially moving further into Europe, would be a focus. "In many cases we may well be able to get there more easily and just as quickly with organic growth, in which case we’ll go down that route. But if the right [acquisition] opportunity comes along we have the ability to take it."
Revenue at Smarts, MSQ’s PR and content agency, grew 15 per cent in 2018 to £6m, according to PRWeek’s UK Top 150 Consultancies list, with 82 staff in offices in Belfast, London, Edinburgh and Glasgow. It clients include Diageo, British Airways, BT and the Scottish Government.
Reid said: "Smarts has always been a success story for us, but for the last two or three years it’s been one of the key engines of growth for the group; in part because it’s a good business, in part because earned media has just become more important in terms of our offer and in terms of the market.
"With LDC support, we are very much looking to try and build on the success of Smarts, and look to raise the profile of the agency as well and look to capitalise on the momentum and the great offer that we think we already have."
He said MSQ "might be that we look to invest more in the brand in the UK and potentially internationally as well", although there are "no specific plans" at present.
MSQ’s other agencies include Holmes & Marchant (branding and design), Stack (customer activation and engagement), Stein IAS (B2B marketing), The Gate (creative and media) and twentysix (digital).
John Clarke, investment director at LDC in Manchester, said: "Peter and the team at MSQ have built a truly formidable business that is able to provide full-service, marketing communications services on an international scale.
"The business retains its independence, agility and a culture that continues to attract some of the best talent in the market. It is this combination that continues to drive MSQ’s growth and we’re looking forward to partnering with the team as we support them with our expertise and significant investment on the next phase of their journey."